On the D(1) extension of the $D(-1)$-Diophantine triples induced by the $D(-1)$-Diophantine pair {1,10}

IF 0.6 3区数学 Q3 MATHEMATICS

Acta Mathematica Hungarica Pub Date : 2025-10-16 DOI:10.1007/s10474-025-01561-z

Y. Briedj, S. E. Rihane

引用次数: 0

Abstract

Let S = {1, 10, c} be an integer set. In this paper, we show that if the product of any two elements of S, decreased by one, forms a perfect square, and if there exists a positive integer d such that $d+1$, $10d+1$ and $cd+1$ are all perfect squares, then $d=d^{\pm}$ where

$$d^{\pm}=19c-11\pm 6\sqrt{(c-1)(10c-1)}.$$

Additionally, under certain conditions, we investigate the integer points on the parametrized family of elliptic curves:

$$E: y^2 = (x+1)(10x+1)(cx+1).$$

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关于$D(-1)$ -Diophantine对诱导的$D(-1)$ -Diophantine三元组的D({1)扩展1,10}

设S = {1,10，c}为整数集。在本文中，我们证明了如果S的任意两个元素的积减去1，形成一个完全平方，并且如果存在一个正整数d使得$d+1$, $10d+1$和$cd+1$都是完全平方，则$d=d^{\pm}$其中$$d^{\pm}=19c-11\pm 6\sqrt{(c-1)(10c-1)}.$$另外，在一定条件下，我们研究了椭圆曲线的参数化族上的整数点：$$E: y^2 = (x+1)(10x+1)(cx+1).$$

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Acta Mathematica Hungarica 数学-数学

CiteScore

1.50

自引率

11.10%

发文量

审稿时长

4-8 weeks

期刊介绍： Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.

On the D(1) extension of the \(D(-1)\)-Diophantine triples induced by the \(D(-1)\)-Diophantine pair {1,10}

Abstract